Semi-classical states of nonlinear Schrödinger equations: a variational reduction method
For a smooth, bounded and uniformly positive potential \(V\), the authors prove the existence of solutions \(v_\varepsilon\) of the nonlinear Schrödinger equation \[ \Delta v_\varepsilon-V(P+\varepsilon y)v_\varepsilon+f(v_\varepsilon)=0, \qquad y\in{\mathbb R}^N, \] vanishing as \(|x|\to\infty\) which approaches a positive ground-state solution \(w\) of the nonlinear Schrödinger equation \[ \Delta w-V(P)w+f(w)=0,\qquad y\in{\mathbb R}^N, \] vanishing at infinity and radially symmetry about some point, under the following hypotheses: (\(f_0\)) \(f(s)\) is of class \(C^1\) and \(f^\prime(s)\) is Lipschitz on \([0,\infty)\), (\(f_1\)) \(\exists p\in(1,{{N+2}\over{N-2}})\) if \(N\geq 3\) such that \(f^\prime(s)=O(s^{p-1})\) as \(s\to+\infty\), (\(f_2\)) \(sf^\prime(s)/f(s)\) is bounded on \((0,1)\), and (\(f_3\)) \(\exists q>1\) such that \(0<qf(s)\leq sf^\prime(s)\) for all \(s>0\). Under these general hypotheses it is left unanswered if the above \(w\) is unique up to translation. A variational method of constructing single and multi-spike solutions, associated with general topologically nontrivial critical points of \(V\), is developed.
scientific article; zbMATH DE number 1837347